Torsion of a linear connection

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This article defines a tensor (viz a section on a tensor bundle over the manifold) of type (1,2)

Definition

Given data

Definition part

The torsion of \nabla, denoted as \tau(\nabla), is defined as a map that takes as input 2 vector fields and outputs a third vector field, as follows:

\tau(\nabla)(X,Y) = \nabla_X Y- \nabla_Y X - [X,Y]

A linear connection whose torsion is zero is termed a torsion-free linear connection.

Facts

Tensoriality

Further information: Torsion is tensorial

The torsion map is a (1,2) tensor. It is tensorial in both X and Y. This means that the value of the torsion of a connection for two vector fields at a point depends only on the values of the vector fields at that point. In other words, \tau(\nabla)(X,Y) at p depends on \nabla, X(p), Y(p) only and does not depend on how X and Y behave elsewhere on the manifold.

Antisymmetry

Further information: Torsion is antisymmetric

We have that the torsion tensor is antisymmetric, i.e., we have:

\tau(\nabla)(Y,X) = -\tau(\nabla)(X,Y)

Equivalently, we have that:

\tau(\nabla)(X,X) = 0